It's a bit ironic that Jaynes didn't fully get it, seeing as he was an early champion of Bayesian thinking.

Indeed. I found Jaynes' book online (the relevant chapter is here), and it's clear that he doesn't get it. Sorry, Missing Shade, but Jaynes clearly dropped the ball on this one.

He writes:

Quote

An incredible amount has been written about this seemingly innocent argument, which leads to an intolerable conclusion.

But the conclusion isn't intolerable -- it's correct, as a Bayesian analysis shows.

He continues:

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But the error in the argument is apparent at once when one examines the equations of probability theory applied to it: the premise, which was not derived from any logical analysis, is not generally true, and he prevents himself from discovering that fact by trying to judge support of an hypothesis without considering any alternatives.

Jaynes is correct that the premise ("an instance of a hypothesis supports the hypothesis") is not always true, but he mistakenly dismisses the entire paradox on this basis. The problem is that in those cases where the premise does hold, the paradox re-emerges.

Jaynes fails to recognize this, and he thus fails to notice that a Bayesian analysis resolves the paradox by affirming it, not by invalidating it.

--------------And the set of natural numbers is also the set that starts at 0 and goes to the largest number. -- Joe G